I've been working on this homework problem for a while now and I can't seem to figure it out.

The problem is as follows:

Prove that for any sets $A$ and $B$, $P(A) \cup P(B) = P(A\cup B)$ then either $A\subseteq B$ or $B\subseteq A$. I'm having trouble understanding what it means to have an equation as one of my hypotheses.

I looked online and I couldn't find anything online, so hopefully this is alright.


Any help would be appreciated


marked as duplicate by Asaf Karagila elementary-set-theory Feb 28 '18 at 23:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Equation as hypothesis: for any real number $x$, if $x^2 = 2$ then either $x = \sqrt{2}$ or $x = -\sqrt{2}$. $\endgroup$ – Patrick Stevens Feb 28 '18 at 22:45
  • $\begingroup$ Probably easiest to prove by contraposition. If neither $A \subseteq B$ or $B \subseteq A$, then there are $x \in A, y \in B$ such that $x \notin B$ and $y \notin A$. Now consider the set $\{x,y\}$. $\endgroup$ – Hans Engler Feb 28 '18 at 22:48
  • $\begingroup$ math.stackexchange.com/questions/246491/… $\endgroup$ – Lucas Corrêa Feb 28 '18 at 22:55